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Financial Innovation

Table 5 The estimation results of EGARCH (1,1,1) model

From: Cue the volatility spillover in the cryptocurrency markets during the COVID-19 pandemic: evidence from DCC-GARCH and wavelet analysis

 

BTC

ETH

XLM

XRP

USDT

ADA

LTC

EOS

Method: ML ARCH – Normal Distribution (OPG—BHHH / Line Search Steps)

\(Mean Equation:d{X}_{t}=\mu +{\o }_{1}d{X}_{t-1}+{\o }_{2}d{X}_{t-2}+{\theta }_{1}{u}_{t-1}+{\theta }_{2}d{u}_{t-2}+{u}_{t}\)

\(\mu\)

36.84

(0.039)

1.186

(0.023)

0.001

(0.240)

0.001

(0.704)

− 2.621

(0.203)

0.000

(0.052)

0.169

(0.177)

0.001

(0.000)

\({\o }_{1}\)

1.816

(0.000)

1.282

(0.000)

1.972

(0.000)

− 1.096

(0.000)

1.078

(0.000)

1.559

(0.000)

1.688

(0.000)

1.124

(0.000)

\({\o }_{2}\)

− 0.913

(0.000)

− 0.947

(0.000)

− 0.975

(0.000)

− 0.945

(0.000)

− 0.353

(0.000)

− 0.921

(0.000)

− 0.939

(0.000)

− 0.164

(0.000)

\({\theta }_{1}\)

− 1.843

(0.000)

− 1.272

(0.000)

− 1.969

(0.000)

1.136

(0.000)

− 1.699

(0.000)

− 1.592

(0.000)

− 1.691

(0.000)

− 1.276

(0.000)

\({\theta }_{2}\)

0.933

(0.000)

0.976

(0.000)

0.971

(0.000)

0.992

(0.000)

0.832

(0.000)

0.979

(0.000)

0.947

(0.000)

0.283

(0.000)

\(Variance Equation={\mathrm{log}(\sigma }_{t}^{2})=\omega +\sum_{j=1}^{q}{\eta }_{j}\mathrm{log}\left({\sigma }_{t-j}^{2}\right)+\sum_{i=1}^{p}{\gamma }_{i}\left|\frac{{\varepsilon }_{t-i}}{{\sigma }_{t-i}}\right|+\sum_{k=1}^{r}{\lambda }_{k}\frac{{\varepsilon }_{t-k}}{{\sigma }_{t-k}}\)

\(\omega\)

− 0.030

(0.581)

− 0.090

(0.000)

− 0.676

(0.000)

− 0.761

(0.000)

− 2.067

(0.000)

− 0.222

(0.000)

− 0.077

(0.000)

− 0.013

(0.188)

\({\eta }_{j}\)

0.113

(0.000)

0.174

(0.000)

0.416

(0.000)

0.495

(0.000)

0.658

(0.000)

0.187

(0.000)

0.142

(0.000)

− 0.083

(0.000)

\({\gamma }_{i}\)

0.043

(0.000)

0.102

(0.000)

0.136

(0.000)

0.129

(0.000)

− 0.215

(0.000)

0.108

(0.000)

0.103

(0.000)

0.156

(0.000)

\({\lambda }_{k}\)

0.997

(0.000)

0.996

(0.000)

0.964

(0.000)

0.954

(0.000)

0.884

(0.000)

0.991

(0.000)

0.992

(0.000)

0.981

(0.000)

\({e}^{\lambda }\)

1.044

1.107

1.146

1.138

0.807

1.114

1.108

1.169

\(AIC\)

14.72

8.022

− 8.125

− 6.195

− 10.01

− 8.085

4.837

− 1.324

\(SIC\)

14.81

8.107

− 8.040

− 6.110

10.93

− 8.000

4.922

− 1.239

\(DW Stat.\)

1.737

1.962

1.747

1.604

2.548

1.783

1.823

1.808

\(Log Likelihood\)

− 3179.1

− 1727.8

1768.0

1350.2

2392.8

1759.5

− 1038.3

295.6

\(ARCH-LM\)

0.254

(0.990)

0.654

(0.767)

1.038

(0.410)

0.591

(0.822)

0.472

(0.908)

1.237

(0.265)

0.355

(0.965)

0.976

(0.464)

  1. p-values are given in parentheses. \(\omega\) is the constant, \(\eta\) is the ARCH effect, \(\gamma\) is the asymmetric effect, and \(\lambda\) is the GARCH effect. Presample variance is selected as backcast with a parameter equal to 0.7. Coefficient covariance is computed using an outer product of gradients (OPG). Error distribution is selected as Gaussian. The optimization method is OPG – Berndt-Hall-Hall-Hausman (BHHH) algorithm with a line search step. Variable X in the mean equation consists of selected cryptocurrencies, respectively used in estimating models. The variables are estimated in their first differences, depending on unit-root results presented in Table 2. ø1 and ø2 show AR(1) and AR(2) coefficients, respectively. θ1 and θ2 show MA(1) and MA(2) coefficients, respectively. μ is the white-noise disturbance term. In consideration of the ARCH – LM test, the heteroskedasticity is tested up to 10 lags
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